Limits and continuity

Transcription

1 CHAPTER 4 Limits and continuity Our first goal is to define and understand lim f(x) =L. Here f : D R where D R. We want the definition to mean roughly, as x gets close to a then f(x) iscloseto L. Perhaps a more careful description is as x gets closer and closer to a with x a, then f(x) gets close to, and could equal, L. What does this mean? It is still vague. It took mathematicians hundreds of years to learn the best way to precisely define this. First note that x has to be able to get arbitrarily close to a without being equal to a. Thus we will need to have a D. It may be that a/ D. If a D then the value f(a) has no affect whatsoever on the limit. Definition. Let D R, a D and L R. Letf : D R. Then lim f(x) =L if for every ε>0thereexistsδ>0sothatifx D, x a <δand x a then f(x) L <ε. (in logical form: ε>0 δ>0 x D, 0< x a <δ f(x) L <ε). We read lim f(x) =L as the limit of f(x) asx approaches a is L Give the logical form of the negation of lim f(x) =L assuming f : D R and a D a) Let f : R R be given by f(x) = 3 for all x R. Let a R. Prove that lim f(x) =3. b) Let f : R R be given by f(x) =x for all x R. Leta R. Prove lim f(x) =a. c) Let f : R R be given by f(x) = 3x +4. Leta R. Prove that lim f(x) = 3a +4. d) Let f : R R be given by f(x) =2x 2 3x + 7. Prove that lim x 1 f(x) =f( 1). Hint: f(x) f( 1) = 2x 2 3x 5 = 2x 5 x +1. Restrictδ 1sotht x ( 1) < δ 1 x ( 2, 0) 2x ( 4, 0) 2x 5 ( 9, 5) 2x 5 < 9. If ε>0isgiven find? so that if δ =min(1,?) then x ( 1) <δ 2x 5 x +1 <ε. e) Let f : R R be given by f(x) = 3x 2 + x 2 and prove that lim x 3 f(x) =f(3). 29

2 30 4. LIMITS AND CONTINUITY 4.3. Let f : R R be given by f(x) = { 0 x is irrational 1 x is rational. Let x R. Prove that lim f(x) does not exist for all a R Let f : D R, a D and L, M R. Assume lim f(x) =L and lim f(x) =M. Prove that L = M Let f : D R and a D. Let L R. Prove that lim f(x) =L for every sequence (a n ) D \{a} with a n a we have f(a n ) L. Hint: For provethe contrapositive State the contrapositive of 4.5. Note. In calculus (408C/D) when you considered limits you required that ε>0so that (a ε, a + ε) \{a} D. Our definition is more general in that we only require a D. In calculus you considered 1-sided limits. Our definition includes that as well. If f :(a, a + ε) R for some ε>0 then lim f(x) =L means in calculus language that lim + f(x) =L or the right hand limit of f(x) ata is equal to L Which of these limits (if any exist)? Prove your answer. a) lim x 0 sin( 1 x ) b) lim x sin 1 x 0 x. 4.8 (Limit Theorems). Let f : D R and g : D R and let a D and L, M R. Assume that lim f(x) =L and lim g(x) =M. Letc R. Provethat a) lim cf(x) =cl. b) lim(f(x)+g(x)) = L + M. c) lim f(x)g(x) =LM. f(x) d) If g(x) 0forx D and M 0 then lim g(x) = L M. Continuous functions. Definition. Let f : D R and let a D. Then f is continuous at a if for all ε>0 there exists δ>0sothatifx D and x a <δthen f(x) f(a) <ε. (in logical form: ε>0 δ>0 x, x D and x a <δ f(x) f(a) <ε) Question: Identify the difference between this and lim f(x) =f(a). If f is continuous at all a D it is called continuous. IfS D and f is continuous at all a S it is called continuous on S.

3 4. LIMITS AND CONTINUITY Let f : R R be given by f(x) = { 1 0 x 1 0 otherwise. At what points is f continuous? In calculus you learned that a continuous function is one whose graph you can draw without taking your pencil off the paper. This is true if the function is continuous on an interval. But it is not an accurate description of continuity at a point or even on the domain D if D is not an interval At which points is f continuous? a) f :[0, 1] R, f(x) =0ifx is irrational and f(x) =1ifx is rational. b) f : R R, f(x) =x if x is rational and f(x) =0ifx is irrational. c) f : R R, f(x) =x 2 + x 1. d) f : D R, f(x) = 1 1 x. D = {x : x 1} Let f : D R and let a D. Prove a) if a is an isolated point of D then f is continuous at a. b) if a D then f is continuous at a if and only if lim f(x) =f(a) let f :[0, 1] {2} R be given by f(x) =x 2 for x [0, 1] {2}. At what points is f continuous? Let f : D R and let a D. Prove f is continuous at a if and only if for every sequence (a n )ind with a n a we have f(a n ) f(a) Give the contrapositive to Let f : D R, g : D R and let a D. Assume that f is continuous at a and g is continuous at a. Letc R. Prove a) f + g is continuous at a. b) cf is continuous at a. c) fg is continuous at a. d) If g(x) 0forx D then f(x) g(x) is continuous at a Prove that every polynomial is continuous on R Let D, E R, f : D E and g : E R. Leta D. Provethatiff is continuous at a and g is continuous at f(a) theng f is continuous at a. Definition. f : D R is bounded if f(d) is bounded. Thus f is bounded K< such that x D, f(x) K.

4 32 4. LIMITS AND CONTINUITY Give an example of a continuous function f :(0, 1] R that is not bounded (a graph will suffice). Can you find a formula for such an f? Let D R be compact and let f : D R be continuous. Prove that f(d) is compact. Hint: Show f(d) is closed and bounded Let D R be compact and let f : D R be continuous. Show a, b D with f(a) f(x) f(b) for all x D. (Thusf takes on a maximum and a minimum value.) Give an example of f :(0, 1) R which is bounded, continuous and has neither a max, nor a min. Can you do the same for f :(0, 1] R? Let f :[a, b] R be continuous and let f(a) < 0 <f(b). Prove that for some x [a, b], f(x) =0. Hint: Let x =sup{y [a, b] :f(y) 0} Let f :[a, b] R be continuous and suppose c lies between f(a) andf(b). Show f(x) =c for some x [a, b] Let f :[a, b] R be continuous. Then f([a, b]) is a closed and bounded interval. Uniform Continuity. If f : D R is continuous on D then a D ε>0 δ>0 x D, x a <δ f(x) f(a) <ε. In general ε depends upon both δ and the point a as previous exercises have illustrated. If we remove the dependence on a we have uniform continuity. Definition. Let f : D R. f is uniformly continuous on D if ε>0 δ>0 x, y D, x y <δ f(x) f(y) <ε Let f : D R be uniformly continuous. Show that f is continuous Let P (x) be a polynomial of degree 1. Prove that P (x) is uniformly continuous on R Negate the definition of uniform continuity Let f(x) =x 2.Provethatf is not uniformly continuous on R Let K R be a compact set and let f : K R be continuous. Prove that f is uniformly continuous. Hint: Let ε>0. x K δ x > 0sothatify K and x y <δ x then f(x) f(y) <ε/2. Choose a finite subcover of K from the open cover {(x δx 2,x+ δx 2 ):x K}. Letδ be the smallest radius in this finite subcover.

5 4. LIMITS AND CONTINUITY Give an example of f : D R which is continuous and such that there exists a Cauchy sequence (x n ) D with (f(x n )) being divergent Let f : D R be uniformly continuous and let (x n ) D be Cauchy. Prove that (f(x n )) is also Cauchy in R.

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